Edgewise Subdivisions, Local $h$-Polynomials, and Excedances in the Wreath Product $\mathbb{Z}_r \wr \mathfrak{S}_n$

Abstract

The coefficients of the local $h$-polynomial of the barycentric subdivision of the simplex with $n$ vertices are known to count derangements in the symmetric group $\mathfrak{S}_n$ by the number of excedances. A generalization of this interpretation is given for the local $h$-polynomial of the $r$th edgewise subdivision of the barycentric subdivision of the simplex. This polynomial is shown to be $\gamma$-nonnegative and a combinatorial interpretation to the corresponding $\gamma$-coefficients is provided. The new combinatorial interpretations involve the notions of flag excedance and descent in the wreath product $\mathbb{Z}_r \wr \mathfrak{S}_n$. A related result on the derangement polynomial for $\mathbb{Z}_r \wr \mathfrak{S}_n$, studied by Chow and Mansour, is also derived from results of Linusson, Shareshian, and Wachs on the homology of Rees products of posets.